BOUNDARY ELEMENT METHODS FOR VIBRATION PROBLEMS

by

Ashok D. Belegundu

Professor of Mechanical Engineering Penn State University

ashokbelegundu@yahoo.com

ASEE Fellow, Summer 2003

Colleague at NASA Goddard: Daniel S. Kaufman Code 542

MY RESEARCH IN LAST FEW YEARS

Noise reduction – passive approaches

V m

Helmholtz Resonator

Stiffeners

TVA/BBVA

Noisy Structure

Dynamic analysis

Acoustic analysis

2

MOST RECENT RESEARCH

Parallel optimization algorithms

05

101520253035

0 16 32 48 64

number of processors

Spee

dup

bracketing

intervalreductiontotal linesearchlinear speedup(reference)

3

OBJECTIVES AT NASA: SUMMER 2003 Tutorial ON Boundary Element Analysis (“BEM”) Computer codes (for tutorial and problem solving) Study the potential for BEM in Vibration at mid-frequency levels i.e. fill the ‘gap’ between FEA and SEA in view of:

Low freq – FEA is very good High freq -- SEA seems adequate (although

detailed response is not obtainable) Assess the state of the art in BEM for vibration analysis

4

BEM -- SIMPLE 1-D EXAMPLE

0)1(,0)0(,02

2

===+ uuxdx

ud

01

02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+∫ dxx

dxudw

Integrate by parts twice:

0''1

0

1

02

21

0

1

0=++− ∫∫ dxxwdx

dxwduwuuw

Choose w such that )(2

2

ixxdx

wd−−= δ

‘Fundamental Solution’ :

)()()( ii xxxxstepxw −−−=

5

SIMPLE EXAMPLE – cont’d Then

∫+−=1

0

1

0

1

0'')( dxxwwuuwxu i (*)

boundary terms domain term

xi = ‘source ’, and w(x) depends on source point xi Choose source xi = 0+ = 0 + ε . Thus w(0) = 0 and remains zero as ε → 0. w(1) = -1, 1)1(' −=w , body force term = -1/3, and we get 3/1)1(' −=uWe can also recover the solution for all interior points from Eq. (*) : u(xi) = (xi – xi

3)/6

Note: the differential (equilibrium) is exactly satisfied within the domain in BEM – while boundary conditions are approximately satisfied, here, the boundary is only wo points, so BEM gives exact solution

6

AXIAL VIBRATION

),(2 txpucu =′′−&&

Harmonic loading: 22 /),( ctxpuku −=+′′

ρω /, Ecc

k ==

( ) 0/0

22 =++′′∫L

dxwcpuku

Integrate by parts twice,

∫∫ ++′′+′−′LL

LL dxcpwdxwkwuwuwu0

2

0

200

/)(

Fundamental solution w :

),(2ixxwkw δ−=+′′

),()(sin1ii xxstepxxk

kw −

−= ⇒

(*)/)(0

200 ∫+′−′=

LLL

i dxcpwwuwuxu

7

F Example

ρ A = 1 du/dx (L) = F/c2 u(0) =0

From Eq. (*), tip displacement, )(tan2 LkckFuL =

This is exact, since boundary conditions are exactly satisfied. Resonances occur when cos (kL) = 0, or kL = π/2, 3π/2, ... Eq. (*) then gives

{ })(sin)(cos)tan()( 2 iii xLkxLkLkckFxu −−−=

Timoshenko’s formula:

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=,...5,3,1 2

2

222

4

12i

L

kL

iLcFu

π

Consider 3 modes in the expansion (FEA needs about 3 elements for each half sine wave) - red color in fig:

8

0 500 1000 15000

0.02

0.04

0.06

0.08

0.1

0.12

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

Mode 4 = blue color

9

BEAM VIBRATION

22

4

4

Acpvk

xv

ρ=−

∂∂

),(24

4

ixxwkxw

δ−=−∂∂

⇒

[ ] [ ]{ })(sin)(sinh2

1),( 2/3 iii xxkxxkk

xxstepw −−−−=

...)( =ixv

But now, also need to differentiate v wrt xi to get additional equation(s) for solution

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STATIC PLATES BY DIRECT BEM

Dpw =∇ 4

Kirchoff theory for thin plates ( ) ( )

terms corner +

+−−=∇−∇ ∫∫∫

****4**4

SA

dSMMwVVwdAwwww θθ

Choose the ‘concentrated force’ fundamental solution w* such that

rrD

w

xxxw

*

ii

ln4

1 giveswhich

),()(

2

*4

π

δ

=

=∇

Thus

( )

( )∑

∫∫∫

=

−+

+−−+=

*K

1k wFFw

dSMMwVVwdAwpD

xwcSA

ii

**

*****

1)( θθ

where ci equals 0.5 if xi is on smooth S, and equals 1 if interior.

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A second equation is obtainable as:

( )( )

( )( )∑

∫∫∫

=

−−+

+−−−+=∂∂

+∂∂

*K

1k i

Si

Aii

wwFFw

dSMMwwVVwdAwpD

wcwc

**

*****

''

'''''1 θθηξ ηξ

where the ‘concentrated moment’ fundamental solution is given by *'w

)cos(ln2

1' απ

rrD

w * =

Expressions for V*, V’* etc are given in the literature. Note: Corners (eg. rectangular plates) require

special attention

Distributed load requires domain integration Here, a function s is defined such that , *2 ws =∇

rrr ∂∂

+∂∂

=∇1

2

22 and then converting to a

contour integral as ∫∫∫ ∂∂

=SA

dSrspdAwp )cos(* β

where β = angle between r and n. Similarly for . *'w

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BEM WITH CONSTANT ELEMENTS : CONTOUR INTEGRATION

element kn

j

W(S) = Wj = constant for e

Gauss point r

source i

Thus, which is evaluated using

gauss quadrature

∫∫ =e

je

dSVwdSwV **

Linear elements will provide more accuracy

Finally: A x = b A is square, unsymmetric, dense, dim 2N+3K where N = no. of regular nodes, K = no of corner nodes Clamped Plate: x = [shear, moment] per unit length Simply-Supported Plate: x = [slope, shear]

13

DYNAMIC ANALYSIS OF PLATES

Dpw

Dhw =+∇ &&

ρ4 For general transient loading, apply Laplace transform and then use BEM. Here, we assume harmonic loading. Thus,

amplitudewewtw ti == )(,)(),( xxx ω

ck

Dpwkw ω

==−∇ ,24

Choose the ‘concentrated force’ fundamental solution w* such that

[ ])()(8

),()()1(

0)2(

0

2*4

rkiHrkHkiw

xxwkxw

*

ii

+=

−=−∇ δ

The ‘concentrated moment’ fundamental solution is given by

*'w

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +)r+=

)(

()()cos('

12

1111

rkKc

kYcrkJciw * α

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CORNER EFFECTS When source point is a corner, two concentrated moment fundamental solutions, and corresponding equations, need to be written. There are a total of three unknowns at each corner When a plate is loaded, corners tend to curl up – corner reactions keep these clamped

Details are omitted here for brevity

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HANDLING DOMAIN (PRESSURE) LOADS IN DYNAMICS

∫∫∫∫AA

dAwpD

dAwpD

** '1,1

Approach 1: Create a domain mesh, and integrate (careful mesh design, and integration are necessary for accuracy) Approach 2: Express , where ai are determined through regression, and each of the known functions ψi can be written as . Then, the Divergence theorem allows us to convert the domain integral to a contour integral.

∑=m

iiawp1

*' ψ

is2∇

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DIFFICULTIES IN BEM

1. Fundamental solutions are hard to derive for complicated differential equations

2. Domain integration requires special care

3. Numerical integration involving bessel functions need special care

17

BEM WITH MULTIPLE RECIPROCITY (BEM-MRM)

Dpwkw =−∇ 24

Choose the ‘concentrated force’ fundamental solution w* such that

),()( of Instead 2*4

ii xxwkxw δ−=−∇

Choose w* as the static fundamental solution :

rrD

w

xxxw

*

ii

ln4

1 giveswhich

),()(

2

*4

π

δ

=

=∇

We then have (ignoring corner terms):

( ) 1

)(

***** ∫∫∫

∫∫

+−−+

+=

Sssss

As

ii

dSMMwVVwdAwpD

xwc

θθ

A

*s

2 dAwwk

Let etcssssws s*

2*

34*

1*

24**

14 ,, =∇=∇=∇

and repeatedly use, with s = si , ( ) ( )∫∫∫ +−−=∇−∇

Ssss

A

dSMMwVVsdAwssw θθ ****4**4

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BEM-MRM (Cont’d) to obtain an expression for the displacement

( )

( )∑

∫∫∫

=

−+

+−−+=

*K

1k wFFw

dSMMwVVwdAwpD

xwcSA

ii

**

*****

ˆˆ

ˆˆˆˆˆ1)( θθ

where

∑∑==

+=+=p

is

is

p

ii

is i

VkVVskww1

*2**

1

*2** ˆ,ˆ etc, along with a similar equation for for the slope.

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Advantages of BEM_MRM Simple fundamental solutions regardless of complexity of differential equation All domain integrals can be easily converted to contour integrals Claimed that number of terms, p, are small for convergence – is this true for all frequencies ? Integrals independent of k can be done once and stored within the frequency loop

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COMPUTER CODE

Circular plate (done), Rectangular plate (in progress) Concentrated load, Constant boundary element Damping: E = E0 (1 + i η) Input: pressure as SPL (converted to point load) Outputs: Center displacement Vs frequency

[ ]

( ) πωω 2/,/w-

,sg'in onacceleratiinc. 1Hzat band,in freqlower & upperand

),(where

,)(1

2

2

nnnn

u

ub

nbb

fgacc(n)

accff

ffN

naccN

PSD

==

==

−=

= ∑

l

l

[ ]∑∞

=

=1

22 )(21

n

naccGRMS

Validation: with Timoshenko’s formulas and with Ansys

21

VALIDATION EXAMPLE

Clamped Circular Steel Plate, R=1m, η=.03 Concentrated load = 1 N

0 20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5x 10

-5

frequency, Hz

Cen

ter D

ispl

acem

ent

Circular Plate

22

ANSYS RESULT

Circular plate concentrated

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SAMPLE PROBLEM Clamped Circular Steel Plate 1m radius, 1 cm thick, η = .01 Pressure corresponds to about 50 Pa 22 1/3-octave bands, ranging from 16Hz to 2000 Hz

0 5 10 15 20 250

50

100

150

SP

L, d

B

SPL in 1/3 Octave Bands

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CENTER DISPLACEMENT Vs HZ

0 500 1000 1500 2000 25000

1

2

3

4

5

6x 10

-3

frequency, Hz

Cen

ter D

ispl

acem

ent

Circular Plate

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MEAN ACCELERATION IN G’S IN EACH

1/3RD OCTAVE BAND

0 5 10 15 20 250

10

20

30

40

50

60

70

80

Cen

ter A

ccel

erat

ion

in g

"s

Circular Plate

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PSD IN EACH 1/3RD OCTAVE BAND

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

g2 /Hz

= in

t(g(f)

2 )/del

taf/d

elta

f

Circular Plate

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DRAWBACK IN MASS MATRIX CALCULATIONS IN FINITE ELEMENTS

u = N q u = displacement field in element e q = nodal displacements of e N = shape functions – polynomials Mass matrix:

∫=e

dVNNm T

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M is assembled from each m Choice of N critical in dynamics, since inertial force equals ω2 M , and solution is obtained from [K - ω2 M]-1 F Comment: for large ω , polynomial shape functions are inadequate Also, shear effects must be properly captured

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A FEW KEY REFERENCES Book: “Boundary elements – an introductory course”, 2nd ed, C.A. Brebbia and J. Dominguez Paper: “Boundary integral equations for bending of thin plates”, Morris Stern Paper: “free and forced vibrations of plates by boundary elements”, C.P. providakis and D.E. Beskos, Computer methods in applied mechanics and engineering, 74, (1989), 23-250 Papers on BEM-MRM: see J. Sladek. Example: “Multiple reciprocity method for harmonic vibration of thin elastic plates”, V. Sladek, J. Sladek, M. Tanaka, Appl. Math. Modelling, 17, (1993), 468-475.

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SUMMARY OF WORK COMPLETED (MAY 19TH – JULY 25TH, 2003)

Tutorial has been developed for BEM in vibration -- rods, beams, Laplace eq., plate bending Computer code for circular plate vibrations

clamped or simply-supported validated with Closed-form solutions and Ansys SPL input, PSD and and other metrics output

State of the art for vibration analysis has been assessed Rectangular (and other geometries) on-going Research issues identified (see next slide)

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FUTURE WORK IN APPLYING BEM FOR PLATE VIBRATIONS

at MID-FREQUENCIES Accurate and efficient numerical integration of body force terms Develop BEM-MRM and compare with ‘standard’ BEM; more validation Include shear effects (BEM-MRM is attractive) Generalize geometries (any planar shape, thick plates, shells) Compare BEM, FEM, Experiment on a specific problem; Spectral element methods Other research: (shock loading, optimization )